A Max-Cut approximation using a graph based MBO scheme

TL;DR

This paper introduces a novel graph-based MBO scheme utilizing a signless Ginzburg-Landau functional for approximating Max-Cut, demonstrating improved accuracy and efficiency over existing methods on certain graph classes.

Contribution

It presents a new approximation algorithm for Max-Cut based on a signless Ginzburg-Landau functional and a graph-based MBO scheme, with proven convergence and superior performance in some cases.

Findings

The proposed method outperforms current state-of-the-art algorithms on some graph classes.

The algorithm has a time complexity of O(|E|), making it efficient for large graphs.

Experimental results show improved Max-Cut approximation accuracy.

Abstract

The Max-Cut problem is a well known combinatorial optimization problem. In this paper we describe a fast approximation method. Given a graph G, we want to find a cut whose size is maximal among all possible cuts. A cut is a partition of the vertex set of G into two disjoint subsets. For an unweighted graph, the size of the cut is the number of edges that have one vertex on either side of the partition; we also consider a weighted version of the problem where each edge contributes a nonnegative weight to the cut. We introduce the signless Ginzburg-Landau functional and prove that this functional Gamma-converges to a Max-Cut objective functional. We approximately minimize this functional using a graph based signless Merriman-Bence-Osher scheme, which uses a signless Laplacian. We show experimentally that on some classes of graphs the resulting algorithm produces more accurate maximum cut approximations than the current state-of-the-art approximation algorithm. One of our methods of minimizing the functional results in an algorithm with a time complexity of O(|E|), where |E| is the total number of edges on G.